reserve x,X,Y for set;
reserve g,r,r1,r2,p,p1,p2 for Real;
reserve R for Subset of REAL;
reserve seq,seq1,seq2,seq3 for Real_Sequence;
reserve Ns for increasing sequence of NAT;
reserve n for Nat;
reserve W for non empty set;
reserve h,h1,h2 for PartFunc of W,REAL;

theorem
  for h being PartFunc of W,REAL, seq being sequence of W holds
  rng seq c= dom h implies abs(h/*seq) = (abs(h))/*seq & -(h/*seq) = (-h )/*seq
proof
  let h be PartFunc of W,REAL, seq be sequence of W;
  assume
A1: rng seq c= dom h;
  then
A2: rng seq c= dom abs(h) by VALUED_1:def 11;
  now
    let n be Element of NAT;
    thus abs(h/*seq).n = |.(h/*seq).n.| by SEQ_1:12
      .= |.h.(seq.n).| by A1,FUNCT_2:108
      .= abs(h).(seq.n) by VALUED_1:18
      .= (abs(h)/*seq).n by A2,FUNCT_2:108;
  end;
  hence abs(h/*seq) = (abs(h))/*seq by FUNCT_2:63;
  thus -(h/*seq) = (-1)(#)(h/*seq) .= ((-1)(#)h)/*seq by A1,Th9
    .= (-h)/*seq;
end;
